Many of the optimal SSP methods, such as SSPERK(5,4) [1]}, [2]}, optimal implicit SSP RK [3]}, and the newly developed optimized SSP IMEX methods in [4]}, were obtained numerically following the methodology in [5]}. Similarly, in constructing an efficient and robust embedded pair for these numerically optimal methods, we construct an analogous optimization problem:
\(F(A, b^T, \tilde{w}) = \begin{bmatrix} \tilde{A}_2^{(p)} &\tilde{A}_\infty ^{(p)} &\left(B_2^{(p+1)} - 1 \right) &\left(B_\infty ^{(p+1)} - 1\right) &\left(C_2^{(p+1)} - 1 \right)&\left(C_\infty ^{(p+1) }- 1 \right)\end{bmatrix}\) 
\(\begin{aligned}& \operatornamewithlimits{arg\,min}_{\tilde{w}}& & \Vert  F(A, b^T, \tilde{w}) \Vert _\infty \\& \text{subject to}& & \tau _k(A, \tilde{w}) = 0, \; k = 1, \ldots , \tilde{p}\\& & & \tilde{w}\ge 0\end{aligned}\) 
